A hovercraft takes off from a platform. Its height (in meters), $x$ seconds after takeoff, is modeled by: $h(x)=-2x^2+20x+48$ How many seconds after takeoff will the hovercraft reach its maximum height?
Answer: The hovercraft's height is modeled by a quadratic function, whose graph is a parabola. The maximum height is reached at the vertex. So in order to find when that happens, we need to find the vertex's $x$ -coordinate. The vertex's $x$ -coordinate is the average of the two zeros, so let's find those first. $\begin{aligned} h(x)&=0 \\\\ -2x^2+20x+48&=0 \\\\ x^2-10x-24&=0 \\\\ (x-12)(x+2)&=0 \\\\ \swarrow &\searrow \\\\ x-12=0\text{ or }&x+2=0 \\\\ x={12}\text{ or }&x={-2} \end{aligned}$ Now let's take the zeros' average: $\dfrac{({12})+({-2})}{2}=\dfrac{10}{2}=5$ In conclusion, the hovercraft will reach its maximum height $5$ seconds after takeoff.